**Assignment 11**

**By: Erica Fletcher**

We will first look at when a and b are equal and k is an integer. Using the equation below we will investigate what happens when we keep **a** and **b** the same and change different values for k (an integer).

Observe when **a = b = k = 1 **in the graph below, we see that the graph is symmetric about the x-axis.

Now let's look at a = b = 1 but k = 2 this time.

Now let's investigate when a = b = 1 but k =3, then k = 5, then k = 8.

We begin to see that as **the value of k** increases, the number of petals on the pictures increases. Lastly as we graph more polar equations we see that the number of leaves is equal to the **k** value for each equation.

Now let's look at **a = b = k = 2 **and **a = b =5 but the value of k stays 2.** What happens as the value of **a** and **b **increases?

So we notice that as the value of **a** and **b **increase the graph gets larger.

Now let's observe when **a > b.**

Examine what happens when **a = 3, b = 1, **and **k = 5.**

In this graph we can see that the number of leaves is still equal to k.

Now let's see what happens when **b > a.**

**Take a = 1, b = 3, and k = 5**.

Even when** b** is greater than

Now let's examine what happens if we add a phase shift. Take a = 1, b = 1, k = 5, and -10 < n < 10.

Click on the following **video** and observe what happens.

Lastly let's see what happens when we replace **cos ( ) with sin ( ).**

Detect what happens when **a = 1, b = 1, and k = 5.**

We notice that our graph looks very similar to the cosine one. The only difference one may notice right away that it is symmetric about the y-axis now as opposed to the x -axis.

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